382
SUPER GRACEFUL LABELING FOR SOME SPECIAL GRAPHS
M.A. Perumal1, S. Navaneethakrishnan2, S. Arockiaraj3 & A. Nagarajan4 1
Department of Mathematics, National Engineering College, K.R.Nagar, Kovilpatti, Tamil Nadu, India. 2,4
Department of Mathematics, V.O.C College, Thoothukudi, Tamil Nadu, India. 3
Department of Mathematics, Mepco Schlenk Engineering College, Sivakasi, Tamil Nadu, India Email :1[email protected], 2[email protected], 3[email protected], 4[email protected]
ABSTRACT
Let G be a (p,q) graph. A bijective function f:V(G) U E(G) →{1,2,...,p+q} such that f(uv)= |f(u)-f(v)| for every edge uvєE(G) is said to be a super graceful labeling. A graph G is called a super graceful graph if it admits a super graceful labeling. In this paper, we show that the graphsPn-1 (1,2,...,n), Coconut tree, Km,n, Sm,n and B(m,n,k) are
super graceful graphs.
Keywords:Graceful labeling, Super graceful labeling and Super graceful graphs.
1. INTRODUCTION
By a graph, we mean a finite undirected graph without loops or multiple edges. A path of length n is denoted by Pn.
A cycle of length n is denoted by Cn . G +
is a graph obtained from the graph G by attaching a pendent vertex to each vertex of G. The concept of graceful labeling has been introduced by Rosa [3] in 1967.
A function f is a graceful labeling of a graph G with p vertices and q edges if f is an injection from the vertices of G to the set {1,2,...,q} such that when each edge uv is assigned the label |f(u) − f(v)| ,the resulting edge labels are distinct. The gracefulness of graphs motivates us to define a new type of labeling, called “ Super graceful labeling”[6].
Let G be a (p, q) - graph. A bijective function f : V (G) ∪ E(G)→ {1, 2, ..., p + q} such that f(uv) = |f(u) − f(v)| for every edge uvє E(G) is said to be a super graceful labeling. A graph G is called a super graceful graph if it admits a super graceful labeling.
In this paper, we show that the graphs Pn−1(1, 2, ..., n), Coconut tree, Km,n, Sm,n and B(m, n, k) are super graceful
graphs.
2. MAIN RESULTS
Definition 2.1. [2] The graph
P
n
1
(
1
,
2
,
3
,...,
n
)
is a graph obtained from a path of vertices v1, v2, ..., vn havingpath length n − 1 by joining i pendent vertices at each of ith vertex. The pendent vertices are labeled as ui,1; ui,2; ...; ui,n for 1 ≤ i ≤ n .
Theorem 2.2. Pn−1(1, 2, ..., n) is a super graceful graph, for n≥ 1 .
Proof. Let G = Pn−1(1, 2, ..., n).
Now |V(G)| = n+n(n+1)/2
,
E(G)=n-1+ n(n+1)/2and
|V (G) U E(G)| = 2n-1+n(n+1)=n2+3n-1. Define f : V (G) ∪ E(G) → {1, 2, ..., n2+ 3n − 1} as follows
2) ( 0 , 1 1,
2 2) (
2) ( 1 , 1 , 1) ( 2 1 1 3 =
) (
2 2
mod i
n i i
i
mod i
n i i
n n
v f i
1
=
)
(
u
1,1f
For
2
i
n
andi
1
(
mod
2)
i j i
j n n u
f ij ( 1) ,1
2 1 2 3 = )
( 2 2
,
For
2
i
n
andi
0
(
mod
2)
i
j
j
i
i
u
f
ij
1
2
,
1
2
1)
(
=
)
383
We construct the vertex labeled sets as follows:
)} ( {
2) (
1 1 = =
1 f vi
mod i
i n
V Let
2
1) ( 1 3
2) (
1 1 = =
2
2 i
n n
mod i
i n
2 1 4 17,..., 3
7, 3 1, 3 =
2 2
2
2 n n
n n n n n n
(or)
2
2
6
17,...,
3
7,
3
1,
3
=
2 2
2
2
n
n
n
n
n
n
n
n
(1)according as
n
is odd or even.)}
(
{
2)
(
0
1
=
=
2
f
v
imod
i
i
n
V
1 2
2) (
2) (
0 1 =
= i i
mod i
i n
2
2
2
.,
3,11,23,..
)
}(
2
3
..,
{3,11,23,.
=
2 2
n
n
or
n
according as
n
is odd or even.{1}
=
)}
(
{
=
1,13
f
u
V
)} ( { 1 = 2) (
0 2 =
= ,
4 f ui j
j i
mod i
i n
V
j
i
n
n
j
i
mod
i
i
n
2
2
1
3
1
=
2)
(
0
2
=
=
2 2
384
2 :1 1
2 1 8 ... 4 1 : 2 7 3 2 1 : 2 1 3 = 2 2 2 n j j n n j j n n j j n n
n j j n n j j n n j j n
n
or 2 :1
2 2 6 ... 4 1 : 2 7 3 2 1 : 2 1 3 = ) ( 2 2 2
according as
n
is odd or even. and
)}
(
{
1
=
2)
(
1
2
=
=
,5
f
u
ijj
i
mod
i
i
n
V
j
i
i
j
i
mod
i
i
n
2
1
2
1)
1)(
(
1
=
2)
(
1
2
=
=
j j j j j n j j n
j 2 :1
2 3 ... 5} 1 : 2 {11 3} 1 : 2 {3 1} = : 1 {2 = 2
)
(
or
2 :1 1
2 2 2 ... 5} 1 : 2 {11 3} 1 : 2 {3 1} = : 1 {2 = 2 n j j n n j j j j j j
according as
n
is odd or even. 2 3 4 4,..., 2 3 2, 2 3 ... 19,21} {13,15,17, {5,7,9} {1} = 2 2 2 n n n n (or)
2
6
2
4,...,
2
2
2
2,
2
2
2
...
19,21}
{13,15,17,
{5,7,9}
{1}
2 2 2n
n
n
n
n
n
according as n is odd or even.
2
3
4
,...,
2
5
,
2
1
21;...;
,15,17,19,
1;5,7,9;13
=
2 2 2n
n
n
n
(or)
2
6
2
,...,
2
6
2
,
2
2
2
21;...,
,15,17,19,
1;5,7,9;13
2 2 2n
n
n
n
n
n
according as
n
is odd or evenWe construct the edge labeled sets as follows:
385
|}
)
(
)
(
{|
2)
(
1
1
=
1
=
1
i i
f
v
v
f
mod
i
i
n
1
2
3)
1)(
(
1)
(
2
1
1
3
2)
(
1
1
=
1
=
n
2n
i
2i
i
mod
i
i
n
2)
3
(
2
4)
6
(2
2)
(
1
1
=
1
=
22
i
i
n
n
mod
i
i
n
|}
2)
3
(
2)
3
(
{|
2)
(
1
1
=
1
=
2
2
i
i
n
n
mod
i
i
n
2}
18,...,4
3
4,
3
{
=
n
2
n
n
2
n
n
(or)
2}
18,...,2
3
4,
3
{
=
n
2
n
n
2
n
n
according as
n
is odd or even.)} ( { 2) (
0 1 =
1
= 1
2
i iv
v f mod i
i n
E
|} ) ( ) ( {| 2) ( 0
1 =
1
= 1
i i f v
v f
mod i
i n
2 2) ( 1 3 1
2 2) (
2) (
0 1 =
1
=
2
2 i
n n i
i
mod i
i n
386
2 3 2
4 4 2
2) (
0 1 =
1
= 2
2 2
n n i
i i i
mod i
i n
|}
2)
3
(
2)
3
(
{|
2)
(
0
1
=
1
=
2
2
n
n
i
i
mod
i
i
n
|} 2) 3 ( 2) 3 ( {| 2) (
0 1 =
1
= 2 2
i i n
n mod
i i n
2}
54,...,2
3
28,
3
10,
3
{
=
n
2
n
n
2
n
n
2
n
n
(or)
2}
54,...,4
3
28,
3
10,
3
{
=
n
2
n
n
2
n
n
2
n
n
according as
n
is odd or even.
)} , ( { 1 = 2) (
1 1 = =
3 f vu j
j i
mod i
i n
E
i i
|} ) ( ) ( {| 1 = 2) ( 1
1 =
= f vi f ui,j
j i
mod i
i n
|})
2
1
2
1)
1)(
(
2
1)
(
1
3
(
{|
1
=
(
2)
(
1
1
=
=
2
2
j
i
i
i
n
n
j
i
mod
i
i
n
j
i
i
i
n
n
j
i
mod
i
i
n
2
1
1
1
2
2
1
1)
3
(
1
=
2)
(
1
1
=
=
2 2 2387 |} 1) ( 2 2 3 {| 1 = 2) ( 1 1 =
= n2 n j i i
j i mod i i n
=2,...,2}
,2
{2
...
38}
3
32,...,
3
30,
3
{
16}
3
14,
3
12,
3
{
2}
3
{
2 2 2 2 2 2 2
n
n
n
n
n
n
n
n
n
n
n
n
n
n
n
n
)
(
or
4}
2,...,2
,4
{4
...
16}
3
14,
3
12,
3
{
2}
3
{
=
n
2
n
n
2
n
n
2
n
n
2
n
n
n
n
according as
n
is odd or even.
) ( 1 = 2) ( 0 1 = = ,4 f viui j
j i mod i i n E
| ) ( ) ( | 1 = 2) ( 0 1 == f vi f ui,j
j i mod i i n
j i n n i i j i mod i i n 2 2 1 3 1 2 2) ( 1 = 2) ( 0 1 = = 2 2
2
3
2
2
2
1
=
2)
(
0
1
=
=
2 2 2n
n
j
i
i
i
j
i
mod
i
i
n
2)
3
(
)
2
1)
(
(
1
=
2)
(
0
1
=
=
i
i
j
n
2n
j
i
mod
i
i
n
) 2 1) ( ( 2) 3 ( 1 = 2) ( 0 1 == n2 n ii j
388
4} 4,...,2 2,4 ,4 {4 ... 26} 3 24, 3 22, 3 20, 3 { 8} 3 6, 3 {
= 2 2 2 2 2 2
n n n n n
n n n n n n n n n n n
)
(
or
=
...
{2
,2
2,...,2
4,...,2}
26}
3
24,
3
22,
3
20,
3
{
8}
3
6,
3
{
2 2 2 2 2 2
n
n
n
n
n
n
n
n
n
n
n
n
n
n
n
according as
n
is odd or even. From the vertex labeled sets and the edge labeled sets, we observe that they are distinct. Their union is{1,2,...,
n
2
3
n
1}
. Therefore,f
is a super graceful labeling and hence,)
(1,2,...,
1
n
P
n is a super graceful graph.Example 2.3 The graphs
P
4(1,2,3,4,5
)
andP
5(1,2,3,4,5
,6)
admit super graceful labeling, such as those shown in Fig.1 and Fig.2
Fig. 1
Fig. 2
Definition 2.4. [2] A graph G = ( V, E ) is called bipartite if
V
=
V
1
V
2
with
=
V
1
V
2
, and every edge of G is of the form
u
,
v
withu
V
1
and
v
V
2
. If each vertex inV
1
is joined with every vertex in2
V
, we have a complete bipartite graph. In this caseV
1
m
and
V
2
n
, the graph is denoted byK
m
,
n
.
Theorem 2.5. Every complete bipartite graph
K
m,n(
m
1,
n
1)
is super graceful.Proof. Let
V
=
V
1
V
2 whereV
1=
{
u
1,
u
2,...,
u
n}
andV
2=
{
v
1,
v
2,...,
v
n}
. Now,|
V
(
K
m,n)
|=
m
n
and|
E
(
K
m,n)
|=
mn
Define
f
:
V
(
K
m,n)
E
(
K
m,n)
{1,2,...,
mn
m
n
}
as follows.m
i
i
u
f
(
i)
=
,
1
andf
(
v
j)
=
m
(
m
1)
j
,
1
j
n
.
389 Let } {1,2,..., = } { 1 = = )} ( { 1 = =
1 i m
i m u f i m V i
and1)}
(
1),...,
2(
1),
(
{
=
}
1)
(
{
1
=
=
)}
(
{
1
=
=
2
m
m
j
m
m
m
m
m
n
m
j
n
v
f
j
n
V
j
We construct the edge labeled set as follows:
)}
(
{
1
=
1
=
=
f
u
iv
jj
n
i
m
E
Let
( )|} ) ( {| 1 = 1 =
= f ui f vj
j n i m
( ( 1) )|} {|
1 = 1 =
= i m m j
j n i m
( 1) }
{ 1 = 1 =
= m m j i
j n i m
}
1)
(
,...,
1)
2(
,
1)
(
{
1
=
=
m
m
i
m
m
i
m
n
m
i
i
m
=}
1)
(
,...,
1)
2(
,
1)
(
{
...
2}
1)
(
2,...,
1)
2(
2,
1)
(
{
1}
1)
(
1,...,
1)
2(
1,
1)
(
{
m
m
n
m
m
m
m
m
m
m
m
n
m
m
m
m
m
m
n
m
m
m
m
m
=1)}.
(
1),...,
1,2(
{
...
2}
1)
(
2,...,
1)
2(
2,
1)
(
{
1}
1)
(
1,...,
1)
2(
1,
1)
(
{
m
n
m
m
m
n
m
m
m
m
m
m
n
m
m
m
m
m
We observe that all the vertex labeled sets and the edge labeled set are distinct and their union is
}
{1,2,...,
mn
m
n
.Therefore,
f
is a super graceful labeling and hence,K
m,n(
m
1,
n
1)
is a super graceful graph.390
Example 2.7 The graph
K
4,5 admits super graceful labeling, such as those shown in Fig.3.Fig.3
Theorem 2.8 Coconut tree is a super graceful graph.
Proof. Let
v
0,
v
1,
v
2,...,
v
i be the vertices of a path, having path lengthi
,
(
i
1)
andv
i1,
v
i2,...,
v
n be the pendent vertices, being adjacent withv
0.Now,
|
V
(
G
)
|=
n
1
and|
E
(
G
)
|=
n
.
Define
f: V(G) U E(G) {1,2,...,2n+1 }as follows.We construct the vertex labeled sets as follows:.
)} ( { 2) ( 0
0 = =
1 f vj
mod j
j i
V Let
1} { 2) ( 0
0 =
=
j mod j
j i
1}
{1,3,...,
)
(
1}
1)
(
{1,3,...,
=
i
or
i
according asi
is odd or even.1}
{1,3,...,
)
(
}
{1,3,...,
=
i
or
i
according asi
is odd or even.0 = = 2
j i
V
according asi
is odd or even.3}
1,...,
1,2
{2
)
(
2}
3,...,
1,2
1,2
{2
=
i
i
i
i
or
i
i
i
according as
i
is odd or even. and1} 5,...,2 3,2
{2 = 1} {2 1 = = )} ( { 1 = =
3 k i i n
i k
n v
f i k
n
V
k
391 )}
( { 2) ( 0
0 =
1
= 1
1
j jv
v f mod j
j i
E
0 =
1
=
j i
0 =
1
= j i
according as
i
is odd or even.4),...,4}
2),2(
,2(
{2
)
(
4),...,2}
2),2(
,2(
{2
=
i
i
i
or
i
i
i
according as
i
is odd or even.)} ( { 2) ( 1
0 =
1
= 1
2
j jv
v f mod j
j i
E
|} ) ( ) ( {| 2) ( 1
0 =
1
= 1
j j f v
v f mod j
j i
|} 1) ( ) 1 (2 {| 2) ( 1
0 =
1
=
j j i mod j
j i
|}
2
2
{|
=
i
j
)} {2( 2) ( 1
0 =
1
= i j
mod j
j i
1))}
(
3),...,2(
1),2(
{2(
)
(
2))}
(
3),...,2(
1),2(
{2(
=
i
i
i
i
or
i
i
i
i
according as
i
is odd or even.3),...,2}
1),2(
{2(
)
(
3),...,4}
1),2(
{2(
=
i
i
or
i
i
according as
i
is odd or even. and)} ( { 1 =
= 0
3 f vv
i k
n
392
|}
)
(
)
(
{|
1
=
=
f
v
f
v
0i
k
n
k
|}
1
1
2
{|
1
=
=
k
i
k
n
} {2 1 =
= k
i k
n
}
2),...,2
1),2(
{2(
=
i
i
n
We observe that all the vertex labeled sets are having odd values and the edge labeled sets are having even values and they distinct. Their union is
{1,2,...,2
n
1}
.Therefore,
f
is a super graceful labeling and hence, coconut tree graph is super graceful.Corollary 2.9 By taking
i
=
1,
in the pathP
i of in the above proof of theorem,K
1,n is a super graceful graph.
Example 2.10 The coconut tree graphs admit super graceful labeling, such as those shown in Fig.4 and Fig.5.
393
Theorem 2.11 The
S
m,n graph is super graceful, form
3
andn
1
.Proof. The
S
m,n graph is obtainedm
paths,n m m m m n n
n
u
u
u
u
u
u
u
u
u
u
u
u
u
u
u
u
,
,
...,
;
...
;
,...
;...
;
0 1 2...
3 2 3 1 3 0 3 2 2 2 1 2 0 2 1 2 1 1 1 0
1 and identify the vertices
0 0 2 0
1
,
u
,...,
u
mu
with
u
0.Now,
|
V
(
S
m,n)
E
(
S
m,n)
|=
(
mn
1)
mn
=
2
mn
1
Define
f
:
V
(
S
m,n)
E
(
S
m,n)
{1,2,...,2
mn
1}
as follows:1
2
=
)
(
=
)
(
00
mn
u
f
u
f
i .For
1
j
n
andj
1
(
mod
2)
m
i
j
i
n
u
f
ji
)
=
2
(
1)
,
1
(
andFor
1
j
n
andj
0
(
mod
2)
m
i
j
i
n
mn
u
f
(
ij)
=
2
1
2
(
1)
,
1
1}
{2
=
)}
(
{
=
)}
(
{
1
=
=
0 01
f
u
f
u
mn
i
m
V
Let
i
)} ( { 2) ( 1
1 = 1 = =
2
j i
u f mod j
j n
i m V
} 1) ( {2
2) ( 1
1 = 1 =
= ni j
mod j
j n
i m
) ( } 1) ( 3,...,2 1) ( 1,2 1) ( {2 1 =
= n i n i n i n or
i m
1} 1)
( 3,...,2 1)
( 1,2 1) ( {2 1 =
= ni ni n i n i
m
according as
n
is odd or even.}
3,...,2
2
1,2
2
;...;2
3,...,3
1,2
;2
{1,3,...,
=
n
n
n
n
mn
n
mn
n
mn
n
)
(
or
1}
3,...,2
2
2
1,
2
1;...;2
3,...,3
1,2
1;2
{1,3,...,
=
n
n
n
n
mn
n
mn
n
mn
n
394
)} ( { 2) ( 0
1 = 1
= =
3
j i
u f mod j
j n
i m V
} 1) ( 2 1 {2 2) ( 0
1 = 1 =
= mn ni j
mod j
j n
i m
1}
1)
(
2
1
4,...,2
1)
(
2
1
2,2
1)
(
2
1
{2
1
=
=
mn
n
i
mn
n
i
mn
n
i
n
i
m
)
(
or
}
1)
(
2
1
4,...,2
1)
(
2
1
2,2
1)
(
2
1
{2
1
=
=
mn
n
i
mn
n
i
mn
n
i
n
i
m
according as
n
is odd or even.2}
3,...,
1,2
2
;
3
2
3,...,2
2
1,2
2
;2
2
3,...,2
1,2
{2
=
mn
mn
mn
n
mn
n
mn
n
mn
n
n
n
n
)
(
or
1}
3,...,
1,2
;2
3
1
3,...,2
2
1,2
2
;2
1
3,...,2
1,2
{2
=
mn
mn
mn
n
mn
n
mn
n
mn
n
n
n
n
according as
n
is odd or even.We construct the edge labeled sets as follows:
)}
(
{
1
=
=
0 11
f
u
iu
ii
m
E
Let
|} ) ( ) ( {| 1 =
= 0 1
i i f u
u f i
m
|} 1) 1) ( (2 1) (2 {| 1 =
= mn ni i
m
|}
1)
(
2
2
{|
1
=
=
mn
n
i
395 1))} ( ( {2 1 =
= n m i i m
}
,...,2
1)
,2(
{2
=
mn
m
n
n
1 22)
(
1
1
=
1
1
=
=
j i j iu
u
f
mod
j
j
n
i
m
E
1 2) ( 1 1 = 1 1 == ij
j i f u
u f mod j j n i m
|} 1)) ( 1) ( 2 1 (2 ) 1) ( (2 {| 1 = 2) ( 1 1 = 1= ni j mn n i j
i m mod j j n
|} 2 2 1) ( 4 {| 1 = 1 = 1 = 2) ( 1 mn j i n i m j n mod j
|} 2 1) ( 4 2 {| 1 = 2) ( 1 1 = 1= mn ni j
i m mod j j n
|} 2 1) ( 4 ,...,2 2 8 ,2 2 4 ,2 2 2 {| 2) ( 1 1 = 1= mn j mn n j mn n j mn n m j
mod j j n
|} ) 1) ( 2 ),...,2( 2 ),2( 2( {| 2) ( 1 1 = 1= mn j mn n j mn n m j
396
)
(
or
=1)}
3),...,2(
2
1),2(
2
{2(
...
1)}
3
3),...,2(
2
1),2(
2
{2(
1)}
3),...,2(
1),2(
{2(
n
mn
n
mn
n
mn
n
mn
n
mn
n
mn
n
mn
mn
mn
according as
n
is odd or even.
1 3 1 = 2) ( 0 1 = 1 = j i j iu u f i m mod j j n E
1
1 = ( 2) ( 0 1 = 1 = j i j i f u
u f i m mod j j n
1) 1) ( (2 ) 1) ( 2 1 (2 1 = 2) ( 0 1 = 1= mn ni j ni j
i m mod j j n
j i n mn i m j n mod j 2 1) ( 4 2 1 = 1 = 1 = 2) ( 0
)} 2 ),...,2( 2 ),2( {2( 2) ( 0 1 = 1= mn j mn n j mn n j
mod j j n
=1)}
4),...,2(
2
2),2(
2
{2(
...
1)}
3
6),...,2(
2
4),2(
2
2),2(
2
{2(
1)}
6),...,2(
4),2(
2),2(
{2(
n
mn
n
mn
n
mn
n
mn
n
mn
n
mn
n
mn
n
mn
mn
mn
mn
)
(
or
=2)}
4),...,2(
2
2),2(
2
{2(
...
2)}
3
4),...,2(
2
2),2(
2
{2(
2)}
6),...,2(
4),2(
2),2(
{2(
n
mn
n
mn
n
mn
n
mn
n
mn
n
mn
n
mn
mn
mn
mn
according as
n
is odd or even.397
Therefore,
f
is a super graceful labeling and hence,S
m,n(
m
3,
n
1)
is a super graceful graph.
Corollary 2.12 By taking
m
=
1
inS
m,n graph, we getS
1,n and it is a super graceful graph. By takingm
=
1
and
n
=
2
inS
m,n we get a path graph and it is a super graceful graph.Example 2.13 The graphs
S
7,6 andS
7,9 admit super graceful labeling, such as those shown in Fig.6 and Fig. 7.Fig. 6
Fig.7
Definition 2.14 The graph
B
(
m
,
n
,
k
)
is a graph obtained from a path of lengthk
by attaching the starK
1,mand
K
1,n with its pendent vertices.Theorem 2.15 The graph
B
(
m
,
n
,
k
)
is a super graceful graph.Proof. Let
u
1,1,
u
1,2,...,
u
1,m be adjacent vertices tov
0 andu
2,1,
u
2,2,...,
u
2,n be another set of adjacent vertices tov
k.Let
v
0 andv
k be terminal vertices of a pathP
k=
v
0v
1v
2...
v
k.
LetG
=
B
(
m
,
n
,
k
)
Now,
|
V
(
G
)
|=
m
n
k
1,
|
E
(
G
)
|=
m
n
k
398
Define
f
:
V
(
G
)
E
(
G
)
{1,2,...,2
(
m
n
k
)
1}
as follows.
2
,
0
,
1(
2)
2)
(
0
1,
0
,
1
)
2(
=
)
(
.
1,1
2
=
)
(
1,m
j
j
k
j
mod
mod
j
k
j
j
k
n
m
v
f
m
i
i
u
f
i j.
1
,
2
2
=
)
(
u
2,m
k
l
l
n
f
l
We construct the vertex labeled sets as follows:
)} ( { 1 =
= 1,
1 f u i
i m
V
Let
1} {2 1 =
= i
i m
1}
,2
{1,3,5,...
=
m
)} ( {
2) ( 0
0 =
1
=
2 f vj
mod j
j k
V
} 1 ) {2(
2) ( 0
0 =
1
= m n k j
mod j
j k
}
1)
1,...,2(
)
1,2(
)
{2(
=
m
n
k
m
n
k
m
n
k
)} ( { 2) ( 1
0 = =
3 f vj
mod j
j k
V
} {2 2) ( 1
0 =
= m j
mod j
j k
and
k
m
m
m
1,2
3,...,2
}
{2
=
)}
(
{
=
2,1 =
4 l
l n
u
f
399
} 2 {2 =
1 =
k l m
l n
}
2
4,...,2
2,2
{2
=
m
k
m
k
m
k
n
We construct the edge labeled sets as follows:
)}
(
{
1
=
=
0 1,1
f
v
u
ii
m
E
Let
|} ) ( ) ( {| 1 =
= f v0 f u1,i i
m
|} 1) (2 1 ) (2(
{| 1 =
= mnk i i
m
|}
1)
(2
1
)
3,...,2(
1
)
1,2(
1
)
2(
{|
=
m
n
k
m
n
k
m
n
k
m
2}
)
2,...,2(
)
),2(
{2(
=
m
n
k
m
n
k
n
k
)} ( { 2) ( 0
0 =
1
= 1
2
j jv
v f mod j
j k
E
|} ) ( ) ( {| 2) ( 0
0 =
1
= 1
j j f v
v f mod j
j k
|} 1) (2
) 1 ) (2(
{| 2) ( 0
0 =
1
=
j m j k n m mod
j j k
|} ) 2(
{| 2) ( 0
0 =
1
= n k j
mod j
j k
1)}
4),...,2(
2),2(
),2(
{2(
400 )}
( {
2) ( 1
1 =
2
= 1
3
j jv
v f mod j
j k
E
|}
)
(
)
(
{|
2)
(
1
1
=
2
=
1
j j
f
v
v
f
mod
j
j
k
|} 1)) ( 1 ) (2(
) (2 {| 2) ( 1
1 =
2
=
j k n m j m mod
j j k
|} )) (2( 2 {| 2) ( 1
1 =
2
= j n k
mod j
j k
)} {2(
2) ( 1
1 =
2
= n k j
mod j
j k
2)}
3),...,2(
1),2(
{2(
=
n
k
n
k
n
)} ( { 1 =
= 2
4 f vkul
l n
E
|} ) ( ) ( {| 1 =
= f vk f u2,l
l n
|} ) 2 1 ) (2( 1) ) (2( {| 1 =
= m n k m n k l
l n
} {2 1 = = l
l n
}
{2,4,...,2
=
n
401 Case ii
k
is evenDefine
f
:
V
(
G
)
E
(
G
)
{1,2,...,2
(
m
n
k
)
1}
as follows..
1
,
2
1
)
2(
=
)
(
2)
(
1
1,
0
,
2
2)
(
0
,
0
,
1
)
2(
=
)
(
1,1
2
=
)
(
1,m
j
j
k
j
mod
f
u
2,m
n
k
l
l
n
mod
j
k
j
j
k
n
m
v
mf
i
i
u
f
i j l
We construct the vertex labeled sets as follows:
)} ( { 0 =
= 1,
'
1 f u i
i m
V
Let
1} {2 0 =
= i
i m
1}
{1,3,...,2
=
m
)} ( {
2) ( 0
0 = =
'
2 f vj
mod j
j k
V
} 1 ) {2(
2) ( 0
0 =
= m n k j
mod j
j k
1}
)
1,...,2(
)
1,2(
)
{2(
=
m
n
k
m
n
k
m
n
k
)} ( { 2) ( 1
0 =
1
=
'
3 f vj
mod j
j k
V
